Finitely Repeated Games 14.12 Game Theory Muhamet Yildiz 1 Road - PDF document

Lecture 12 Finitely Repeated Games 14.12 Game Theory Muhamet Yildiz 1

Road Map 1. Entry-Deterrence/Chain-store paradox 2. Finitely repeated Prisoners Dilemma 3. A general result 4. Repeated games with multiple equilibria 2

Prisoners' Dilemma, repeated twice, many times • Two dates T = {O,I}; • At each date the prisoners' dilemma is played: C D C 5,5 0,6 D 6,0 1,1 • At the beginning of I players observe the strategies at o. Payoffs= sum of stage payoffs. 3

Twice-repeated PD I c o 2 c c D D 1 1 o o 105116 5061 11 6 12 7 6 1 7 2 10 11 5 6 11 12 6 7 56016712 What would happen ifT = {O, I ,2, ... , n}? 4

Microsoft v. a Startup Startup NP SE u u NU o 2 1 1 1 2 3 4 5

Microsoft v. Startups su NP SE M S u U / MS sur SU MS M M U NU j :: NU j U NU j :: NU j : NU 0 2 2 4 2 2 3 3 3 1 3 2 2 2 3 6 5 6 8 4 5 3 4 5 7 4 5 6 7 What wo uld hap pe n i ft here are n startups? 6

Entry deterrence 2 1 Enter Acc. ... ... (1 , 1) X Fight , , , , (0,2) (-1,-1) 7

~ ~ ~ Entry deterrence, repeated twice 1 Frier 2 A::c. \ Frier 2 A::c. (2,2) :x HI#: :x Fil#: A::c. 2 Frier \ ,3) ~ (1 ,3) (0 , 0) Frier 2 A::c. :x HI#: (0 , 0) :x Fil#: ( -1 ,1 ) (0,4) (- \,\) (-2, - 2) 8

A general result • G = "stage game" = a finite game ,n} • T = {O,l, ... • At each t in T, G is played, and players remember which actions taken before t; • Payoffs = Sum of payoffs in the stage game. • Call this game G(T). Theorem: If G has a unique subgame-perfect equilibrium s*, G(T) has a unique subgame- perfect equilibrium, in which s* is played at each stage. 9

With multiple equilibria T={O,l} s* = oAt t = 0, play (B,M) 2 oAt t = I, play (C,R) if (B,M) at t L M R I = 0, play (A,L) otherwise. A 1,1 5,0 0,0 B 0,5 0,0 M 4,4 L R 0,0 0,0 A 6,1 1,1 C 3,3 2,2 B 1,6 1 ,1 7,7 1,1 C 1,1 4,4 10

Can you see on the path ofSPE? T= {O ,l} L M R • (B,M) (B,M) 5,0 0,0 A 1,1 • (B,M) (A,L) 0,5 0,0 B 4,4 • (B,L) (C,R) 0,0 0,0 C 3,3 • (C,L)(C,R) • Take T={0,1 ,2} • (C,L) (B,M) (C,R) 11

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